Number Series & Patterns
Spotting the rule in a sequence and continuing it.
This page is a sequence, not a document: try a question cold, learn the methods, drill until they’re automatic, then prove it under a clock. Skipping straight to reading is the least effective thing you can do here — that’s not opinion, it’s the testing effect.
Try one first
Before any teaching — a real question tells you honestly where you stand.
What number comes next? 10, 14, 18, 22, 26, …
The methods
Every number series & patterns pattern the tests use — the method in a few lines, then a worked example.
Arithmetic sequence
Check the first differences: if they're constant, add the difference once more for the next term.
Worked example
What number continues the series? 4, 11, 18, 25, __
Answer: 32
First differences are a constant +7, so the next term is 25 + 7 = 32. (uniqueness: only the arithmetic (+7) rule fits.)
Geometric sequence
Check the ratio between consecutive terms: if it's constant, multiply the last term by it.
Worked example
What number continues the series? 3, 6, 12, 24, __
Answer: 48
Each term doubles (×2), so the next is 24 × 2 = 48. The 36 foil treats the last gap (+12) as additive. (uniqueness: geometric ×2 fits; arithmetic does not.)
Quadratic sequence
If first differences aren't constant, check the second differences; extend the pattern of differences to get the next term.
Worked example
What number continues the series? 2, 5, 10, 17, 26, __
Answer: 37
First differences are 3, 5, 7, 9 — rising by 2 each time. The next difference is 11, so 26 + 11 = 37 (the series is n²+1). (uniqueness: constant second difference; neither arithmetic nor geometric fits.)
Alternating sequence
Test the odd- and even-position terms as two separate series; continue the one the next position belongs to.
Worked example
What number continues the series? 3, 20, 6, 17, 9, 14, 12, __
Answer: 11
Two interleaved series: positions 1,3,5,7 go 3,6,9,12 (+3); positions 2,4,6 go 20,17,14 (−3). The next term is in the −3 thread: 14 − 3 = 11. (uniqueness: no single arithmetic/geometric/quadratic rule fits — the interleaved split is required.)
Operation cycle
Look for a repeating cycle of operations (e.g. ×3 then −2) and apply the next operation in the cycle.
Worked example
What number continues the series? 2, 6, 4, 12, 10, 30, 28, __
Answer: 84
The operations cycle ×3 then −2: 2→6→4→12→10→30→28. The next operation is ×3, so 28 × 3 = 84. (uniqueness: no constant difference/ratio/2nd-difference fits — the ×3,−2 cycle is required.)
Drill it
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Prove it under a clock
A skill isn't test-ready until it survives time pressure, mixed with everything else.
Your mastery map on the dashboard tracks this area from every answer you give — including the ones on this page.